3.1320 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=184 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c^2 d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(7*c*d*(b*d + 2*c*d*x)^(7/2)) + (2*Sqrt[a + b*x + c*x^2])
/(21*c*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/
(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d
])], -1])/(21*c^2*(b^2 - 4*a*c)^(3/4)*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.435064, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c^2 d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(9/2),x]

[Out]

-Sqrt[a + b*x + c*x^2]/(7*c*d*(b*d + 2*c*d*x)^(7/2)) + (2*Sqrt[a + b*x + c*x^2])
/(21*c*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2)) + (Sqrt[-((c*(a + b*x + c*x^2))/
(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d
])], -1])/(21*c^2*(b^2 - 4*a*c)^(3/4)*d^(9/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 95.0312, size = 170, normalized size = 0.92 \[ - \frac{\sqrt{a + b x + c x^{2}}}{7 c d \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{2 \sqrt{a + b x + c x^{2}}}{21 c d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{21 c^{2} d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(9/2),x)

[Out]

-sqrt(a + b*x + c*x**2)/(7*c*d*(b*d + 2*c*d*x)**(7/2)) + 2*sqrt(a + b*x + c*x**2
)/(21*c*d**3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)) + sqrt(c*(a + b*x + c*x**2)
/(4*a*c - b**2))*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(
1/4))), -1)/(21*c**2*d**(9/2)*(-4*a*c + b**2)**(3/4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.664262, size = 175, normalized size = 0.95 \[ \frac{c (b+2 c x) (a+x (b+c x)) \left (4 c \left (3 a+2 c x^2\right )-b^2+8 b c x\right )+\frac{i (b+2 c x)^{11/2} \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}}}}{21 c^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x + c*x^2]/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(c*(b + 2*c*x)*(a + x*(b + c*x))*(-b^2 + 8*b*c*x + 4*c*(3*a + 2*c*x^2)) + (I*(b
+ 2*c*x)^(11/2)*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sq
rt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/Sqrt[-Sqrt[b^2 - 4*a*c]])/(21*c^2*
(b^2 - 4*a*c)*(d*(b + 2*c*x))^(9/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.064, size = 659, normalized size = 3.6 \[ -{\frac{1}{42\,{d}^{5} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{4}{c}^{2}} \left ( 8\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{3}{c}^{3}+12\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{x}^{2}b{c}^{2}+6\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}x{b}^{2}c+\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{1 \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{1 \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{3}+16\,{c}^{4}{x}^{4}+32\,b{c}^{3}{x}^{3}+40\,{x}^{2}a{c}^{3}+14\,{x}^{2}{b}^{2}{c}^{2}+40\,xab{c}^{2}-2\,{b}^{3}cx+24\,{a}^{2}{c}^{2}-2\,ac{b}^{2} \right ) \sqrt{d \left ( 2\,cx+b \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(9/2),x)

[Out]

-1/42*(8*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4
*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*
EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^
(1/2))*(-4*a*c+b^2)^(1/2)*x^3*c^3+12*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^
(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2)
)/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+
b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*b*c^2+6*((b+2*c*x+(-4*
a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*
((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/
2)*x*b^2*c+((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(
-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),
2^(1/2))*(-4*a*c+b^2)^(1/2)*b^3+16*c^4*x^4+32*b*c^3*x^3+40*x^2*a*c^3+14*x^2*b^2*
c^2+40*x*a*b*c^2-2*b^3*c*x+24*a^2*c^2-2*a*c*b^2)/d^5*(d*(2*c*x+b))^(1/2)/(c*x^2+
b*x+a)^(1/2)/(4*a*c-b^2)/(2*c*x+b)^4/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}}{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(9/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x + a)/((16*c^4*d^4*x^4 + 32*b*c^3*d^4*x^3 + 24*b^2*c^2*
d^4*x^2 + 8*b^3*c*d^4*x + b^4*d^4)*sqrt(2*c*d*x + b*d)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(1/2)/(2*c*d*x+b*d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x + a)/(2*c*d*x + b*d)^(9/2), x)